How to Predict Nesting Sites Analysis of the Problem and How to - PowerPoint PPT Presentation

Importance of . . . Importance of . . . Studying Spatial . . . Predicting Nesting . . . How to Predict Nesting Sites Analysis of the Problem and How to Measure How Can We Solve . . . How Can We Gauge . . . Shoreline Erosion: Fuzzy and How to Measure . . . Resulting Algorithm: . . . Probabilistic Techniques Home Page for Environment-Related Title Page Spatial Data Processing ◭◭ ◮◮ ◭ ◮ Stephen M. Escarzaga, Craig Tweedie, Page 1 of 20 Olga Kosheleva, and Vladik Kreinovich Go Back University of Texas at El Paso, El Paso, TX 79968, USA smescarzaga@utep.edu, ctweedie@utep.edu, Full Screen olgak@utep.edu, vladik@utep.edu Close Quit

Importance of . . . Importance of . . . 1. Importance of Environment-Related Spatial Studying Spatial . . . Data Processing Predicting Nesting . . . • When analyzing the ecological systems, it is important Analysis of the Problem to study: How Can We Solve . . . How Can We Gauge . . . – the spatial environment of these systems, and How to Measure . . . – spatial distribution of the corresponding species in Resulting Algorithm: . . . this spatial environment. Home Page • In most locations within an ecological zone, the envi- Title Page ronmental changes are reasonably slow. ◭◭ ◮◮ • It usually takes decades to see a drastic change. ◭ ◮ • However, at the borders between different ecological Page 2 of 20 zones, the changes are much faster. Go Back Full Screen Close Quit

Importance of . . . Importance of . . . 2. Importance of Studying Shorelines Studying Spatial . . . • In the border between different types of plants the Predicting Nesting . . . changes are fast but still gradual: Analysis of the Problem How Can We Solve . . . – new types of plants appear; their proportion grows, How Can We Gauge . . . and eventually, How to Measure . . . – they take over the area. Resulting Algorithm: . . . • However, there are border areas where the change is Home Page the most drastic: namely, the shorelines. Title Page • The shorelines are, in most places, retreating because ◭◭ ◮◮ of the shoreline erosion. ◭ ◮ • The overall area of the shorelines is relatively small. Page 3 of 20 • However, they are a habitat for many species, from Go Back birds (like seagulls) to turtles. Full Screen • From this viewpoint, it is important to be able to trace and measure shoreline erosion. Close Quit

Importance of . . . Importance of . . . 3. Studying Spatial Distribution of Different Studying Spatial . . . Species Predicting Nesting . . . • It is important to trace and measure spatial environ- Analysis of the Problem ments which are important for different species. How Can We Solve . . . How Can We Gauge . . . • It is also necessary to trace spatial location of these How to Measure . . . species. Resulting Algorithm: . . . • This problem is especially important for rare birds. Home Page • Birds are most vulnerable when they at their nesting Title Page sites. ◭◭ ◮◮ • It is therefore important to monitor these sites. ◭ ◮ • Some species use the same nesting sites year after year. Page 4 of 20 • Birds from other species vary their sites each year. Go Back • To be able to monitor birds from these species, it is Full Screen important to be able to predict their nesting sites. Close Quit

Importance of . . . Importance of . . . 4. Predicting Nesting Sites: Formulation of the Studying Spatial . . . Problem Predicting Nesting . . . • We observe nesting sites for a certain bird species. Analysis of the Problem How Can We Solve . . . • Our goals are: How Can We Gauge . . . – to analyze which criteria are important for selecting How to Measure . . . nesting sites, and Resulting Algorithm: . . . – to come up with formulas that would enable us to Home Page predict nesting sites. Title Page • Let v 1 , . . . , v n be parameters that may influence the ◭◭ ◮◮ selection of a nesting site. ◭ ◮ • Examples: parameters describing elevation, hydrology, Page 5 of 20 vegetation level, distance form other nesting sites, etc. Go Back • For each geographical location x , we record the values Full Screen of these parameters v 1 ( x ), . . . , v n ( x ). Close Quit

Importance of . . . Importance of . . . 5. Formulation of the Problem (cont-d) Studying Spatial . . . • We assume that the birds select a nesting site based on Predicting Nesting . . . the values of these quantities (at least some of them). Analysis of the Problem How Can We Solve . . . • So, a bird tries to maximize the value of some objective How Can We Gauge . . . function F ( v 1 , . . . , v n ) depending on v i . How to Measure . . . • We do not know the exact form of the dependence Resulting Algorithm: . . . F ( v 1 , . . . , v n ). Home Page • However, we can expand F in Taylor series and keep Title Page the first few terms up in this expansion. ◭◭ ◮◮ • If we only keep linear terms, we get: ◭ ◮ n � F ( v 1 , . . . , v n ) = a 0 + a i · v i . Page 6 of 20 i =1 Go Back • If we also keep quadratic terms, we get: Full Screen n n n � � � F ( v 1 , . . . , v n ) = a 0 + a i · v i + a iℓ · v i · v ℓ , Close i =1 i =1 ℓ =1 Quit

Importance of . . . Importance of . . . 6. Formulation of the Problem (cont-d) Studying Spatial . . . • For each of these approximations, the (unknown) ob- Predicting Nesting . . . jective function has the form Analysis of the Problem N How Can We Solve . . . � F ( v 1 , . . . , v n ) = A j · V j ( x ) , where: How Can We Gauge . . . j =1 How to Measure . . . • V j ( x ) are known values (e.g., v i ( x ) and v i ( x ) · v ℓ ( x )); Resulting Algorithm: . . . • A j are the coefficients that need to be determined. Home Page • We assume that each year, each of the observed nesting Title Page sites x k : ◭◭ ◮◮ – has the largest possible value of the objective func- ◭ ◮ tion Page 7 of 20 – in the set C k of all locations x which are closer to Go Back x k that to any other nesting locations. Full Screen • Under this assumption, we want to find A 1 , . . . , A N that best explain the observations. Close Quit

Importance of . . . Importance of . . . 7. Analysis of the Problem Studying Spatial . . . • The fact that on the cell C j , the linear function (2.1) Predicting Nesting . . . attains its largest value at the site x j means that Analysis of the Problem How Can We Solve . . . N N � � A j · V j ( x k ) ≥ A j · V j ( x ) for all x ∈ C k . How Can We Gauge . . . How to Measure . . . j =1 j =1 Resulting Algorithm: . . . • In other words, we should have Home Page N � Title Page def A · ∆( x ) = A j · ∆ j ( x k ) ≥ 0 , where ◭◭ ◮◮ j =1 ◭ ◮ def def A = ( A 1 , . . . , A n ) , ∆( x ) = (∆ 1 ( x ) , . . . , ∆ N ( x )) , Page 8 of 20 def and ∆ j ( x ) = V j ( x k ) − V j ( x ) . Go Back • Similarly, we should have A · ( − ∆( x )) ≤ 0 for all x . Full Screen Close Quit

Importance of . . . Importance of . . . 8. How Can We Solve This Problem? Studying Spatial . . . • From the mathematical viewpoint, this problem is sim- Predicting Nesting . . . ilar to the linear discriminant analysis , when: Analysis of the Problem – we have two sets S and S ′ and How Can We Solve . . . How Can We Gauge . . . – we need to find a hyperplane that separates them, How to Measure . . . i.e., a vector A such that A · S ≥ 0 for all S ∈ S and A · S ′ ≤ 0 for all S ′ ∈ S ′ . Resulting Algorithm: . . . Home Page • In our case, S is the set of all vectors ∆ j ( x ), and S ′ is Title Page the set of all vectors − ∆ j ( x ). ◭◭ ◮◮ • The standard way of solving this problem is to com- ◭ ◮ pute: Page 9 of 20 – the mean µ of all M vectors S ∈ S , Go Back – the covariance matrix Σ, and Full Screen – then to take A = Σ − 1 µ . Close Quit

Importance of . . . Importance of . . . 9. How to Solve the Problem Studying Spatial . . . • In our case, we should do the following: Predicting Nesting . . . Analysis of the Problem – compute all M vectors ∆( x ) with components How Can We Solve . . . ∆ j ( x ) = V j ( x k ) − V j ( x ), where x ∈ C k ; � How Can We Gauge . . . – compute the average µ = 1 ∆( x ); M · How to Measure . . . x Resulting Algorithm: . . . – compute the corresponding covariance matrix: Home Page � Σ ab = 1 (∆ a ( x ) − µ a ) · (∆ b ( x ) − µ b ); M · Title Page x ◭◭ ◮◮ – compute the desired weights as A = Σ − 1 µ . ◭ ◮ • We can predict the nesting locations as the points x at Page 10 of 20 � N which A j · V j ( x ) is the largest. Go Back i =1 Full Screen • Instead of the above probabilistic clustering, we can use fuzzy clustering . Close Quit

Importance of . . . Importance of . . . 10. How Can We Gauge the Accuracy of the Re- Studying Spatial . . . sulting Estimate Predicting Nesting . . . • To gauge the accuracy of this prediction, we can test Analysis of the Problem it against the observed data. How Can We Solve . . . How Can We Gauge . . . • For each cell C k , we compute the location c k at which � N How to Measure . . . F = A j · V j ( x ) is the largest in this cell. Resulting Algorithm: . . . i =1 Home Page • As a natural measure of prediction accuracy, we can Title Page take the mean square distance between: ◭◭ ◮◮ – these predicted nesting sites c k and ◭ ◮ – the actual nesting sites x k . Page 11 of 20 Go Back Full Screen Close Quit